Question

A bookseller has $420$ science stream books and $130$ Arts stream books. He wants to stack them in such a way that each stack has the same number and they take up the least area of the surface.

Answer 1

Hint: We need to find a way to stack a given set of books such that each stack should have the same number and they take up the least area of the surface. For this find the HCF of $420$ and
$130$ . The result will be the maximum number of books that a stack can have.

Complete step by step answer:
We need to find a way to stack a given set of books such that each stack should have the same number and they take up the least area of the surface.
It is given that there are $420$ science stream books and $130$ in Arts stream books.
Let us find the Highest Common Factor (HCF) of these.
For this, let us first find the LCM of each numbers.
\[\begin{align}
  & 2\left| \!{\underline {\,
  420 \,}} \right. \\
 & 2\left| \!{\underline {\,
  210 \,}} \right. \\
 & 3\left| \!{\underline {\,
  105 \,}} \right. \\
 & 5\left| \!{\underline {\,
  35 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
LCM of $420=2\times 2\times 3\times 5\times 7$
$\begin{align}
  & 2\text{ }\left| \!{\underline {\,
  130 \,}} \right. \\
 & 5\text{ }\left| \!{\underline {\,
  65 \,}} \right. \\
 & 13\left| \!{\underline {\,
  13 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
LCM of $130=2\times 5\times 13$
The common factors of these are $2\times 5=10$
Hence, $HCF(420,120)=10$ .
Hence, each stack can have $10$ books.


Note:
 The greatest number which divides each of the two or more numbers is called Highest Common Factor (HCF) , Greatest Common Measure(GCM) or Greatest Common Divisor(GCD). Do not get confused with HCF and LCM. LCM or Least Common Multiple is used to find the smallest common multiple of any two or more numbers. HCF of a number can also be determined by writing down the factors and taking the highest common of them. For example,
Factors of $420$ are:$420=1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6,\text{ }7,~10,\text{ }12,\text{ }14,\text{ }15,\text{ }20,\text{ }21,\text{ }28,\text{ }30,\text{ }35,\text{ }42,\text{ }60,\text{ }~70,\text{ }84,\text{ }105,\text{ }140,\text{ }210,\text{ }420~$ Factors of $130$ are:
$130=1,\text{ }2,\text{ }5,~10,\text{ }13,\text{ }26,\text{ }65,\text{ }130$
Of these, the common factors are $1,2,5,10.$
The highest among these is $10$ .
Hence, the $HCF(420,120)=10$